Publicação
On the supercritical KDV equation with time-oscillating nonlinearity
| Resumo: | For the initial value problem (IVP) associated to the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, \begin{equation*} u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, \end{equation*} numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003) and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, \end{equation*} where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large. |
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| Autores principais: | Panthee, Mahendra |
| Outros Autores: | Scialom, Marcia |
| Assunto: | Korteweg-de vries equation Cauchy problem Local and global well-posedness |
| Ano: | 2013 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| Resumo: | For the initial value problem (IVP) associated to the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, \begin{equation*} u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, \end{equation*} numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003) and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, \end{equation*} where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large. |
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